Convergence of excursion point processes and its applications to functional limit theorems of Markov processes on a half-line

Invariance principles are obtained for a Markov process on a half-line with continuous paths on the interior. The domains of attraction of the two different types of self-similar processes are investigated. Our approach is to establish convergence of excursion point processes, which is based on It\^{o}'s excursion theory and a recent result on convergence of excursion measures by Fitzsimmons and the present author.